Chapter 7, Problem 65RE

To determine

To find: The exact value of the expression $\cos \frac{y}{2}$ if $\sec x=\frac{3}{2}$ and $\csc y=3$ where $x$ and $y$ are in Quadrant I.

Answer to Problem 65RE

The value of the expression $\cos \frac{y}{2}$ is $\pm \sqrt{\frac{3+2\sqrt{2}}{6}}$ .

Explanation of Solution

Given information:

The given expression is $\cos \frac{y}{2}$ . The values are $\sec x=\frac{3}{2}$ and $\csc y=3$ .

Calculation:

The values are $\sec x=\frac{3}{2}$ and $\csc y=3$ . Since $x$ and $y$ are in Quadrant I. All values will be positive.

  $\begin{array}{l}\sec x=\frac{3}{2}\\ \cos x=\frac{2}{3}\end{array}$

Apply Pythagoras theorem to calculate the value of perpendicular as $\sqrt{5}$ . So, the value of $\sin x$ will be $\frac{\sqrt{5}}{3}$ .

  $\begin{array}{c}\csc y=3\\ \sin y=\frac{1}{3}\end{array}$

Apply Pythagoras theorem to calculate the value of base as $\sqrt{8}$ . So, the value of $\cos y$ will be $\frac{\sqrt{8}}{3}$ .

Solve the expression using sum formula.

  $\begin{array}{c}\cos \frac{y}{2}=\pm \sqrt{\frac{1+\cos y}{2}}\\ =\pm \sqrt{\frac{1+\frac{\sqrt{8}}{3}}{2}}\\ =\pm \sqrt{\frac{3+\sqrt{8}}{6}}\\ =\pm \sqrt{\frac{3+2\sqrt{2}}{6}}\end{array}$

Therefore, the value of the expression $\cos \frac{y}{2}$ is $\pm \sqrt{\frac{3+2\sqrt{2}}{6}}$ .